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I am assuming the decimal digits are 1 zero then 1 one then 2 zeros then 2 ones then 3 zeros then 3 ones etc..

Suppose it has period $n$ after some point $\omega$, then you can find a string of $n$ 1s in a row, and a string of $n$ 0s in a row: That contradicts periodicity because every string of length n should be the same after $\omega$.

For example if it has period 3 after 20 letters, you can keep going a bit further to find